Solving simultaneous equations
It is common to work with problems involving the intersection of lines. To find the point of intersection when linear functions are expressed in the form $y=f(x)$y=f(x), set the equations equal to each other and solve for $x$x.
What if the equations are in general form, or there are more than two equations? In these cases, the problem becomes that of solving a system of simultaneous equations.
Two methods may be used to do this: solution by substitution or by elimination. Note that the number of equations being solved must be equal to or greater than the number of variables to allow for the full solution of the system.
It is a good idea to check that any solutions found work in all equations to ensure that the set is consistent. This is particularly important for systems of equations with more equations than unknowns. These types of systems are called over-determined.
Solution by substitution
If one of the equations can easily be solved for one of the variables, this result can be substituted in place of that variable in any other equation/s. After simplifying, solve for the remaining variable.
Worked example
Find the intersection of the graphs of the two linear equations $3y+4x=1$3y+4x=1 and $-2y+x=3$−2y+x=3.
Think: Can one of the variables be isolated easily?
Do:
The second equation can be solved for $x$x easily since $x=3+2y$x=3+2y.
This identity can then be substituted in place of $x$x in the first equation:
| $3y+4\left(3+2y\right)$3y+4(3+2y) | $=$= | $1$1 |
| $3y+12+8y$3y+12+8y | $=$= | $1$1 |
| $11y$11y | $=$= | $-11$−11 |
| $y$y | $=$= | $-1$−1 |
This solution is then substituted back into the second equation to give $x$x. Hence $x=3+2\times\left(-1\right)$x=3+2×(−1) and $x=1$x=1.
Solution by elimination (addition method)
Using algebra to eliminate a variable from a system of equations will make it easier to solve. This is done by multiplying (or dividing) equations by constants to make the coefficients of a particular variable the same magnitude in each (signs may be different). The equations can then be added or subtracted to eliminate this variable, and the new equation solved for the remaining unknown.
Worked example
Find the intersection of the graphs of the two linear equations $3y+4x=1$3y+4x=1 and $-2y+x=3$−2y+x=3.
Think: Can each equation be multiplied or divided by some amount to give the same coefficient for one of the variables?
Do:
| Equation 1: | $3y+4x$3y+4x | $=$= | $1$1 | (x2) |
| Equation 2: | $-2y+x$−2y+x | $=$= | $3$3 | (x3) |
| Equation 1: | $6y+8x$6y+8x | $=$= | $2$2 |
| Equation 2: | $-6y+3x$−6y+3x | $=$= | $9$9 |
Add these equations together to eliminate $y$y and give:
$11x=11$11x=11
So, $x=1$x=1. This identity can then be substituted in place of $x$x into either equation. The second equation gives $-2y+1=3$−2y+1=3 which implies that $y=-1$y=−1. This is the same solution as found using the substitution method.
Simultaneous equations may be solved by hand using the methods of substitution or elimination.
Technology, such as numerical solvers, may also be used.
Practice questions
QUESTION 1
We want to solve the following system of equations using the substitution method.
| Equation 1 | $3x+4y=4$3x+4y=4 |
| Equation 2 | $2x+5y=-2$2x+5y=−2 |
First solve for $x$x.
Hence, solve for $y$y.
QUESTION 2
Use the elimination method by adding both equations to solve for $y$y and then for $x$x.
| Equation 1 | $8x+3y=-11$8x+3y=−11 |
| Equation 2 | $-8x-5y=29$−8x−5y=29 |
First solve for $y$y
Now solve for $x$x
Question 3
How to use the CASIO Classpad to complete the following tasks regarding simultaneous equations
Consider the following system of equations:
| Equation 1 | $y=x-3$y=x−3 |
| Equation 2 | $2x+5y=20$2x+5y=20 |
Solve the system of linear equations using the graphing functionality of your CAS calculator, leaving your answer as a pair of coordinates.
Solve the system of linear equations using the solving functionality of your CAS calculator, leaving your answer as a pair of coordinates.
Question 4
How to use the TI Nspire to complete the following tasks regarding simultaneous equations
Consider the following system of equations:
| Equation 1 | $y=x-3$y=x−3 |
| Equation 2 | $2x+5y=20$2x+5y=20 |
Solve the system of linear equations using the graphing functionality of your CAS calculator, leaving your answer as a pair of coordinates.
Solve the system of linear equations using the solving functionality of your CAS calculator, leaving your answer as a pair of coordinates.